Consider the blending system shown in Fig. $1.$ A feedback control system is used to reduce the effect of disturbances in feed composition x$1$ on the controlled variable, product composition, x$.$ Inlet flow rate w$2$ can be manipulated. Do the following: $($a$)$ Draw a block diagram of the feedback control system. $($b$)$ Using the information shown below, derive a transfer function for each block. $($c$)$ Simulate the closed$-$loop response for the PI controller se$$ings given below and a step disturbance of $+0\mathrm{.}2$ in x$1.($d$)$ Repeat part $($c$)$ for a set$-$point change of $0\mathrm{.}1.$ A$$empt to obtain be$$er closed$-$loop responses by tuning the PI controller. Which controller se$$ings give the best results? $($e$)$ A$$empt to obtain improved control by adding derivative action to your best PI controller of part $($d$).$ Try several values of derivative time, $\backslash$tau D$.$ Which one gives the best results? $($f$)$ Suppose that the sampling line to the composition analyzer becomes partially plugged so that the measurement time delay is now three minutes. Using your best controller se$$ings of part $($d$),$ simulate the closed$-$loop response for the same setpoint change and the new time$-$delay value. Explain your new simulation results. Does the larger time delay have a major effect on control system performance?Consider the blending system shown in Fig. $1.$ A feedback control system is used to

reduce the effect of disturbances in feed composition $x1$ on the controlled variable,

product composition, $x.$ Inlet flow rate $w2$ can be manipulated. Do the following:

$($a$)$ Draw a block diagram of the feedback control system.

$($b$)$ Using the information shown below, derive a transfer function for each block.

$($c$)$ Simulate the closed$-$loop response for the PI controller settings given below and

a step disturbance of $+0\mathrm{.}2$ in $1.$

$($d$)$ Repeat part $($c$)$ for a set$-$point change of $-0\mathrm{.}1.$ Attempt to obtain better closed$-$loop

responses by tuning the PI controller. Which controller settings give the best results?

$($e$)$ Attempt to obtain improved control by adding derivative action to your best PI

controller of part $($d$).$ Try several values of derivative time, ${}_D.$ Which one gives the

best results?

$($f$)$ Suppose that the sampling line to the composition analyzer becomes partially

plugged so that the measurement time delay is now three minutes. Using your best

controller settings of part $($d$),$ simulate the closed$-$loop response for the same set$-$

point change and the new time$-$delay value. Explain your new simulation results.

Does the larger time delay have a major effect on control system performance?

Process Information The

pilot$-$scale blending tank

has an internal diameter

of $2m$ and a height of $3m.$

Inlet flow rate $w1$ and

inlet composition $2$ are

constant. The nominal

steady$-$state operating

conditions are as follows:

$\frac{?}{\text{b}\text{a}\text{r}}(w{)}_1=650k\frac{g}{m}in{\stackrel{}{x}}_1=0\mathrm{.}2=1\mathrm{.}5m\frac{?}{b}$

$ar(w{)}_2=350k\frac{g}{m}in{\stackrel{}{x}}_2=0\mathrm{.}6$

$=1\frac{g}{c}{m}^3\stackrel{}{x}=0\mathrm{.}34$